An iteration due to artist mathematician [1] Barry Martin produces intricate doiley-like patterns by iterating a simple mathematical function. I ran across this via [2].
The images produced are sensitive to small changes in the starting parameters x and y, as well as to the parameters a, b, and c.
Here are three examples:
And here’s the Python code that was used to make these plots.
import matplotlib.pyplot as plt
from numpy import sign, empty
def make_plot(x, y, a, b, c, filename, N=20000):
xs = empty(N)
ys = empty(N)
for n in range(N):
x, y = y - sign(x)*abs(b*x - c)**0.5, a - x
xs[n] = x
ys[n] = y
plt.scatter(xs, ys, c='k', marker='.', s = 1, alpha=0.5)
plt.axes().set_aspect(1)
plt.axis('off')
plt.savefig(filename)
plt.close()
make_plot(5, 5, 30.5, 2.5, 2.5, "doiley1.png")
make_plot(5, 6, 30, 3, 3, "doiley2.png")
make_plot(3, 4, 5, 6, 7, "doiley3.png")
[1] Tom Crilly identified Martin as an artist in [2] and I repeated this error. Barry Martin informed me of the error in the comments below.
[2] Desert Island Theorems: My Magnificent Seven by Tony Crilly. The Mathematical Gazette, Mar., 2001, Vol. 85, No. 502 (Mar., 2001), pp. 2-12
If you’re interested in Hopalong attractors like this, see: https://examples.pyviz.org/attractors
Those are reminiscent of a Poincare section of a KAM torus. I suspect they’re the same phenomenon at some level. The KAM torus arises generically by a nonlinear perturbation of a periodic motion. In this case the underlying periodic motion is found when a=b=c=0.
If you google Barry Martin you will find that the Hopalong iteration is not due to ‘The Artist’ Barry Martin but to me the mathematician Barry Martin. I must have created a lot of custom for the artist, but I don’t mind that! Tony Crilly has long since apologized for the error. My discovery was pure serendipity when I was researching the Henon attractor. I retired in 1998 am now 84 and still besotted by maths. Hopalong continues to provide surprises.